A pair of vectors \((w,z)\)  satisfying (REF ) is called a solution of the LCP\((q,M)\) . A vector \(z\)  is called a \(z\) -solution if there exists a vector \(w\)  such that \((w,z)\)  is a solution of the LCP\((q,M)\) . Similarly vector \(w\)  is called a \(w\) -solution if there exists a vector \(z\)  such that \((w,z)\)  is a solution of the LCP\((q,M)\) . A \(w\) -solution, \(\tilde{w},\)  of the LCP\((q, M)\)  is said to be locally \(w\) -unique if there exists a neighborhood of \(\tilde{w}\)  within which \(\tilde{w}\)  is the only \(w\) -solution.
Xu [1]} introduced column competent matrices. Dutta et al. [2]} studied column competent matrices in context of linear complementarity problem. Xu [1]} showed that LCP\((q,A)\)  has locally unique \(w\) -solution if and only if \(A\)  is column competent matrix. Several matrix classes and their subclasses have been studied extensively because of their predominance in scientific computing, complexity theory, and the theoretical foundations of the linear complementarity problems. For details see [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. The problem of computing the value vector and optimal stationary strategies are formulated as a linear complementary problem for these two classes of undiscounted zero-sum games. This provides an alternative proof of the orderfield property for these two classes of games. For details see [12]}, [13]}, [14]}, [15]}, [16]}, [17]}. We can write down formulations of QMOP as LCP related weighted problem. For details see [18]}. The applicability of Lemke’s algorithm and the concept of principal pivot transform extend the class of LCP problems solvable by Lemke’s algorithm. For details see [19]}, [20]} [21]}, [22]}, [23]}, [8]}, [9]}, [26]}.
